Challenging +1.2 This is a standard Further Maths skew lines question requiring systematic application of perpendicularity conditions (dot products = 0) to find two parameters, then a follow-up distance calculation. While it involves multiple steps and algebraic manipulation, the method is algorithmic and well-practiced in FP1, making it moderately above average difficulty but not requiring novel insight.
The line \(l _ { 1 }\) passes through the point \(A\) whose position vector is \(4 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }\) and is parallel to the vector \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\). The line \(l _ { 2 }\) passes through the point \(B\) whose position vector is \(\mathbf { i } + 7 \mathbf { j } + 11 \mathbf { k }\) and is parallel to the vector \(\mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k }\). The points \(P\) on \(l _ { 1 }\) and \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vectors of \(P\) and \(Q\).
Find the shortest distance between the line through \(A\) and \(B\) and the line through \(P\) and \(Q\), giving your answer correct to 3 significant figures.
The line $l _ { 1 }$ passes through the point $A$ whose position vector is $4 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }$ and is parallel to the vector $3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$. The line $l _ { 2 }$ passes through the point $B$ whose position vector is $\mathbf { i } + 7 \mathbf { j } + 11 \mathbf { k }$ and is parallel to the vector $\mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k }$. The points $P$ on $l _ { 1 }$ and $Q$ on $l _ { 2 }$ are such that $P Q$ is perpendicular to both $l _ { 1 }$ and $l _ { 2 }$. Find the position vectors of $P$ and $Q$.
Find the shortest distance between the line through $A$ and $B$ and the line through $P$ and $Q$, giving your answer correct to 3 significant figures.
\hfill \mbox{\textit{CAIE FP1 2013 Q11 EITHER}}